Finite Group Databasebeta

← #29662624

Subgroups — #29662624

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456246C211yesmaximal, characteristic
2#21455849C212yesmaximal, complemented
3#446×D61C211yesmaximal, characteristic, complemented
4#21445314C211yesmaximal, characteristic, complemented
5#21445313C211yesmaximal, characteristic, complemented
6#21445300C211yesmaximal, characteristic, complemented
7#21445290C212yesmaximal, complemented
8#21445289C212yesmaximal, complemented
9#21444564C211yesmaximal, characteristic, complemented
10#21444465C212yesmaximal, complemented
11C61×#1688C211yesmaximal, characteristic, complemented
12C2×#14519796—21no—
13C2×#14519796—21no—
14#16840310C2212yes—
15C4×Q8×D61C2211yescharacteristic
16#16840145—21no—
17#16840145—21no—
18#16840142C2211yescharacteristic
19#16840134C2212yes—
20#16840079—22no—
21#16840048C2212yes—
22#16840047C2212yes—
23#16839959C2212yes—
24#16839899—22no—
25#16839893C2211yescharacteristic
26#16839892C2211yescharacteristic
27#16839891—21no—
28#16839890—21no—
29#16839889C2211yescharacteristic
30#16839853—22no—
31SD16×Dic61—22no—
32#16839814C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35#97×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839514C2212yes—
38#16839513C2211yescharacteristic
39#16839509C2212yes—
40#16839507C2212yes—
41#16839506C2212yes—
42C61×#493C2211yescharacteristic
43C61×#460C2212yes—
44C61×#446C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#331C2212yes—
47C2×#11758224—22no—
48C2×Q8×D61D411yescharacteristic
49#14519796—44no—
50#14519796—44no—
51C2×Dic244—21no—
52C2×Dic244—21no—
53C2×#11758200—22no—
54C2×#11758200—22no—
55C2×#11758200—22no—
56C2×C8×D61—21no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519777C2312yes—
60#14519764—22no—
61#14519748—22no—
62#14519748—22no—
63Q8×Dic61D411yescharacteristic
64C4×Dic122—21no—
65C4×Dic122C2311yescharacteristic
66#14519742—21no—
67#14519742D411yescharacteristic
68D4×Dic61—22no—
69D4×Dic61—22no—
70C4⋊C4×D61—21no—
71C4⋊C4×D61D411yescharacteristic
72C42×D61—21no—
73C42×D61C2311yescharacteristic
74#14519709—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2311yescharacteristic
79#14519706—21no—
80#14519706—21no—
81#14519706—21no—
82#14519702—22no—
83#14519701—22no—
84#14519698—22no—
85#14519689—22no—
86#14519688—21no—
87#14519687—21no—
88#14519687—21no—
89C4×#92418C2311yescharacteristic
90C8×Dic61—22no—
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×SD16×C61—22no—
94C2×D8×C61—21no—
95C2×D8×C61—21no—
96#127×C61C2312yes—
97C4×Q8×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2312yes—
101#102×C61—22no—
102#97×C61—21no—
103C4×C8×C61C2311yescharacteristic
104C2×Q8×C61C2×D411yescharacteristic
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×C22⋊D61—42no—
109C22×Dic61—42no—
110Q8×D61—42no—
111#11758224—44no—
112#11758224—44no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61#4111yescharacteristic
118C2×Dic122—21no—
119C2×Dic122—22no—
120C2×Dic122C2×D411yescharacteristic
121SD16×C61—44no—
122D8×C61—42no—
123D8×C61—42no—
124C2×C8×C61—21no—
125C2×C8×C61—22no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C4⋊C4×C61—21no—
129C4⋊C4×C61C2×D411yescharacteristic
130C42×C61—21no—
131C42×C61C2411yescharacteristic
132C22⋊Dic61—42no—
133C22⋊Dic61—42no—
134#11758200—44no—
135#11758200—44no—
136#11758200—44no—
137C4⋊Dic61—21no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61C2×D411yescharacteristic
148C4×Dic61C2×D411yescharacteristic
149C4×Dic61#4111yescharacteristic
150C2×#92418—22no—
151C2×#92418—21no—
152C2×#92418C2×D411yescharacteristic
153C2×#92418C2×D411yescharacteristic
154Dic244—42no—
155Dic244—42no—
156C8×D61—22no—
157C8×D61—42no—
158C8×D61SD1614yescomplemented
159C23×C61—42no—
160C22×D61#12111yescharacteristic
161Q8×C61—22no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—42no—
165D4×C61—42no—
166D4×C61—82no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C22⋊D61—84no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—42no—
176C2×Dic61—84no—
177C2×Dic61#12111yescharacteristic
178C2×Dic61#12111yescharacteristic
179C2×Dic61#12111yescharacteristic
180C4×D61—22no—
181C4×D61—41no—
182C4×D61—42no—
183C4×D61—42no—
184C4×D61C2×SD1612yes—
185C4×D61AGL(1,ℤ/8)12yes—
186Dic122—22no—
187Dic122—42no—
188Dic122—42no—
189Dic122—42no—
190C8×C61—21no—
191C8×C61—41no—
192C8×C61—42no—
193C8×C61C2×SD1612yes—
194#92418—21no—
195#92418—41no—
196#92418—42no—
197#92418C2×SD1612yes—
198#19146—611nomaximal, Sylow
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61C2×#12111yescharacteristic
203C2×D61—21no—
204C2×D61—21no—
205C2×D61#46012yes—
206C4×C61—21no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61C22×SD1611yescharacteristic
210C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C4×C61#57512yes—
212Dic61—21no—
213Dic61—21no—
214Dic61—41no—
215Dic61—41no—
216Dic61—82no—
217Dic61#46012yes—
218#3147—611no—
219#3103—612no—
220C2×#446—611no—
221#1690—611no—
222#1690—611no—
223#1688—611no—
224#1688D6111yescharacteristic, complemented
225#1686—612no—
226#1686—612no—
227#1564—611no—
228#1540—612no—
229C2×C61—162no—
230C2×C61C2×#46011yescharacteristic
231C2×C61#305411yescharacteristic
232C2×C61#306711yescharacteristic
233D61—22no—
234C2×#135—1221no—
235C2×#135—1221no—
236#532—612no—
237C2×C4×Q8—611no—
238#495—1221no—
239#495—1221no—
240#494—611no—
241#493C2×D6111yescharacteristic
242#492—612no—
243#481—1222no—
244#475—612no—
245#475—612no—
246#460—612no—
247#460C2×D6112yes—
248#448—1222no—
249#446—611no—
250#446—611no—
251#446—611no—
252#446—1221no—
253#446—1221no—
254#446C2×D6111yescharacteristic
255C4×SD16—1222no—
256C4×SD16—1222no—
257#431—612no—
258C2×C4⋊C8—611no—
259C2×C4⋊C8—1221no—
260C2×#97—1221no—
261C2×C4×C8—611no—
262#335—612no—
263#334—611no—
264#334C2×D6111yescharacteristic
265#332—612no—
266#332—612no—
267#331—612no—
268#331C2×D6112yes—
269C61#1914611yescharacteristic, complemented, Sylow
270C2×#41—1222no—
271C22×Q8—611no—
272#135—2444no—
273#135—2444no—
274C2×Q16—1221no—
275C2×Q16—1221no—
276C2×SD16—22no—
277C2×SD16—1222no—
278C2×SD16—1222no—
279C2×SD16—1222no—
280C2×D8—21no—
281C2×D8—21no—
282C22×C8—611no—
283C22×C8—611no—
284C22×C8—1221no—
285C4⋊Q8—612no—
286#127C22×D6112yes—
287#124—1222no—
288#121—1222no—
289#121—1222no—
290C4×Q8—611no—
291C4×Q8—611no—
292C4×Q8—611no—
293C4×Q8—1221no—
294C4×Q8—1221no—
295C4×Q8C22×D6111yescharacteristic
296C4×D4—1222no—
297C4×D4—1222no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—1221no—
300C2×C42—611no—
301C2×C42—1221no—
302#106—1222no—
303#106—1222no—
304C4⋊C8—21no—
305C4⋊C8—611no—
306C4⋊C8—612no—
307C4⋊C8—1221no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310C4⋊C8—1221no—
311C4⋊C8C22×D6111yescharacteristic
312C4⋊C8C22×D6112yes—
313#103—1222no—
314#103—1222no—
315#102—22no—
316#102—1222no—
317C22⋊C8—1222no—
318#97—21no—
319#97—1221no—
320#97—1221no—
321#97—1221no—
322C4×C8—611no—
323C4×C8—611no—
324C4×C8—611no—
325C4×C8—1222no—
326C4×C8C22×D6111yescharacteristic
327#41—2444no—
328#41—2444no—
329C2×Q8—611no—
330C2×Q8—1221no—
331C2×Q8—1222no—
332C2×Q8—2442no—
333C2×Q8D4×D6111yescharacteristic
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—2442no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—2442no—
344Q16—2442no—
345Q16—2442no—
346SD16—44no—
347SD16—2444no—
348SD16—2444no—
349SD16—2444no—
350D8—42no—
351D8—42no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—611no—
355C2×C8—614no—
356C2×C8—611no—
357C2×C8—1222no—
358C2×C8—1222no—
359C2×C8—1221no—
360C2×C8—2442no—
361C2×C8D4×D6111yescharacteristic
362C2×C8D4×D6111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—611no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4D4×D6111yescharacteristic
372C22⋊C4—2442no—
373C22⋊C4—2442no—
374C42—21no—
375C42—611no—
376C42—611no—
377C42—611no—
378C42—1221no—
379C42—1221no—
380C42—1221no—
381C42C23×D6111yescharacteristic
382C23—42no—
383C23—611no—
384Q8—22no—
385Q8—1222no—
386Q8—2442no—
387Q8—2442no—
388Q8—2442no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—82no—
394D4—4884no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—611no—
398C2×C4—612no—
399C2×C4—612no—
400C2×C4—611no—
401C2×C4—611no—
402C2×C4—1222no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—4884no—
406C2×C4—2441no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4C2×D4×D6111yescharacteristic
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4#41×D6111yescharacteristic
413C8—21no—
414C8—41no—
415C8—42no—
416C8—612no—
417C8—1221no—
418C8—2441no—
419C8—2442no—
420C8SD16×D6112yes—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—612no—
425C22—1221no—
426C22—1221no—
427C22#121×D6111yescharacteristic
428C4—21no—
429C4—41no—
430C4—41no—
431C4—612no—
432C4—1221no—
433C4—1221no—
434C4—4882no—
435C4—2441no—
436C4—2441no—
437C4C2×SD16×D6111yescharacteristic
438C4AGL(1,ℤ/8)×D6111yescharacteristic
439C4#1684055812yes—
440C2—162no—
441C2—1222no—
442C2#460×D6111yescharacteristic
443C2#2145542611yescharacteristic
444C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1688C2
socleC22×C61C2×#121
prime radical#21456246C2
hypercenter#1688D61
hypocenterC61#19146
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1688

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21455849C2semidirect
3. #446×D61C2semidirect
4. #21445314C2semidirect
5. #21445313C2semidirect
6. #21445300C2semidirect
7. #21445290C2semidirect
8. #21445289C2semidirect
9. #21444564C2semidirect
10. #21444465C2semidirect
11. C61×#1688C2semidirect
158. C8×D61SD16semidirect
224. #1688D61semidirect
269. C61#19146semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214562462
#214558492
#446×D612
#214453142
#214453132
#214453002
#214452902
#214452892
#214445642
#214444652
C61×#16882
#1914661